Well, you’d think the answer would be nice and easy, but it isn’t.
An antenna does not need to be plugged into a wall socket, except when it does. That doesn’t help a lot, does it?
Let’s explain. An antenna itself is just a metal stick. It doesn’t need power to function. What an antenna is doing is actually receiving a broadcast from the air and converting that broadcast into a very weak electrical signal that flows through the antenna and into the cable. If you’re using a coaxial cable, a very tiny amount of that signal is lost in converting from the 300 ohm impedance of the antenna to the 75 ohm impedance of the coaxial cable. But otherwise, there’s no external power needed.
Unless… you have an amplified antenna. Amplified antennas do need power from a wall socket but it’s not the antenna that needs power, it’s the amplifier. The whole purpose of an amplifier is to read a weak signal, add power from an external source (like the wall) and create a stronger signal. So an amplified antenna is always going to need power. Most amplified antennas won’t work without a power source because the amplifier circuits are built right into the antenna so the signal just gets stopped if there’s no power to the amplifier.
If you have an amplified antenna, you also need to make sure the cable running to it has a solid copper center conductor. Most “home store” cables don’t… their cables have copper plating over a zinc center. That kind of cable won’t carry the electricity that the amplifier needs.